Related papers: Log-concavity of the overpartition function

We prove that the partition function $p(n)$ is log-concave for all $n>25$. We then extend the results to resolve two related conjectures by Chen. The proofs are based on Lehmer's estimates on the remainders of the Hardy--Ramanujan and the…

Let $\bar{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher order $\log$-concavity property of the overpatition function in a similar framework done by Hou and Zhang for the partition function. This…

Let $\overline{p}(n)$ denote the overpartition funtion. This paper presents the $2$-$\log$-concavity property of $\overline{p}(n)$ by considering a more general inequality of the following form \begin{equation*} \begin{vmatrix}…

Let $\overline{p}(n)$ denote the overpartition function. In this paper, we obtain an inequality for the sequence $\Delta^{2}\log \ \sqrt[n-1]{\overline{p}(n-1)/(n-1)^{\alpha}}$ which states that \begin{equation*} \log…

In the $1970$s Nicolas proved that the coefficients $p_d(n)$ defined by the generating function \begin{equation*} \sum_{n=0}^{\infty} p_d(n) \, q^n = \prod_{n=1}^{\infty} \left( 1- q^n\right)^{-n^{d-1}} \end{equation*} are log-concave for…

The partition function $p(n)$ and many of its related restricted partition functions have recently been shown independently to satisfy log-concavity: $p(n)^2 \geq p(n-1)p(n+1)$ for $n\geq 26$, and satisfy the inequality: $p(n)p(m) \geq…

Bessenrodt and Ono, Chen, Wang and Jia, DeSalvo and Pak were the first to discover the log-subadditivity, log-concavity, and the third-order Tur\'{a}n inequality of partition function, respectively. Many other important partition statistics…

In recent years, there has been extensive work on inequalities among partition functions. In particular, Nicolas, and independently DeSalvo--Pak, proved that the partition function $p(n)$ is eventually log-concave. Inspired by this and…

In the $1970$s, Nicolas proved that the partition function $p(n)$ is log-concave for $ n > 25$. In \cite{HNT21}, a precise conjecture on the log-concavity for the plane partition function $\func{pp}(n)$ for $n >11$ was stated. This was…

Let $\mathcal{A}=(a_i)_{i=1}^\infty$ be a non-decreasing sequence of positive integers and let $k\in\mathbb{N}_+$ be fixed. The function $p_\mathcal{A}(n,k)$ counts the number of partitions of $n$ with parts in the multiset…

We prove that a general class of measures, which includes $\log$-concave measures, is $\frac{1}{n}$-concave according to the terminology of Borell, with additional assumptions on the measures or on the sets, such as symmetries. This…

In this paper, we prove that the number of unimodal sequences of size $n$ is log-concave. These are coefficients of a mixed false modular form and have a Rademacher-type exact formula due to recent work of the second author and Nazaroglu on…

Euler's gamma function is logarithmically convex on positive semi-axis. Additivity of logarithmic convexity implies that the function sum of gammas with non-negative coefficients is also log-convex. In this paper we investigate the series…

We prove that the number $q(n)$ of partitions into distinct parts is log-concave for $n \geq 33$ and satisfies the higher order Tur\'an inequalities for $n\geq 121$ conjectured by Craig and Pun. In doing so, we establish explicit error…

Let $p(n)$ denote the partition function. Desalvo and Pak proved the log-concavity of $p(n)$ for $n>25$ and the inequality $\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)>\frac{p(n)}{p(n+1)}$ for $n>1$. Let $r(n)=\sqrt[n]{p(n)/n}$ and…

It is shown that, if nu >= 1/2 then the generalized Marcum Q function Q_nu(a, b) is log-concave in 0<=b <infty. This proves a conjecture of Sun, Baricz and Zhou (2010). We also point out relevant results in the statistics literature.

We combinatorially prove that the number $R(n,k)$ of permutations of length $n$ having $k$ runs is a log-concave sequence in $k$, for all $n$. We also give a new combinatorial proof for the log-concavity of the Eulerian numbers.

In 1918, Hardy and Ramanujan made a breakthrough by developing the circle method to deduce an asymptotic formula for the partition function $p(n)$, which was later refined by Rademacher in 1937 to produce an absolutely convergent series…

Menon's proof of the preservation of log-concavity of sequences under convolution becomes simpler when adapted to 2-sided infinite sequences. Under assumption of log-concavity of two 2-sided infinite sequences, the existence of the…

Heim, Neuhauser, and Tr\"oger recently established some inequalities for MacMahon's plane partition function $\mathrm{PL}(n)$ that generalize known results for Euler's partition function $p(n)$. They also conjectured that $\mathrm{PL}(n)$…